Component Relationships and Requirements

o[language] A model is the model for one and
only one ontological language. Models are required to declared their
associated language, either by indexing it or embedding it. The same comment
applies to ontologies. The embedding of a language within an ontology can be
either concentrated in one location or distributed throughout the ontology.

o[satisfaction] An ontology is a collection of
expressions of the associated language that are regarded as assertions or
constraints. The (meta) assertion that a model satisfies an expression
(or more generally, an ontology) can only be made when both have the same
associated language. When a model is asserted to satisfy an ontology, but has
not declared a language, the language of the ontology is implicitly declared
for the model.

o[truth context] An ontological language
defines a (formal) context called the truth context. This context is a
(meta) classification, whose instances are models for the language, whose
types are expressions of the language, and whose classification relation is
satisfaction between models and expressions.

o[type equivalence] Ordinary assertions appear
in ontologies, not languages. However, synonymic type equivalence is a
special assertion that appears in languages, and is inherited by their
ontologies and models. In a language the assertions of synonymic relation
type equivalence must be compatible with the assertions of synonymic entity
type equivalence.

o[language extension] Along with language
inclusion, synonymic type equivalence defines morphisms of languages. The
assertion that language1
extends language0
means that all type symbol declarations and type equivalence assertions made
in language0
are type symbols and type equivalences of language1; but that there
may be further type symbol declarations and type equivalence assertions made
in language1.
Language extension is a morphism of languages.

o[truth infomorphism] A language morphism from
language0
to language1
defines a (meta) infomorphism from truth0
to truth1,
consisting of an expression map from expression(language0) to
expression(language1)
and a (functorial) passage from model(language1) to model(language0).

o[ontology extension] The (meta) assertion that
ontology1
extends ontology0
means language1
extends language0
and the satisfaction closure of ontology1
includes the satisfaction closure of ontology0.